One great 'Luke Skywalker as a baby fanfiction' could be about his early days on Tatooine being secretly protected by Obi - Wan. Maybe it focuses on how Obi - Wan made sure Luke was safe from the Empire's reach while also trying to figure out how to train him in the ways of the Force when he grew up.
There might be a fanfiction where Luke as a baby is discovered by some local Tatooine families who are in awe of his 'strange' powers that seem to manifest even at that young age. This could lead to a story of them trying to protect him from those who would do harm in exchange for a reward from the Empire.
A possible 'Luke Skywalker as a baby fanfiction' could be set during the time when his mother was still alive. It could explore the relationship between Padmé, Anakin, and baby Luke, and how they tried to create a future for him in a galaxy that was starting to crumble under the dark shadow of the Sith.